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Advanced Derivatives and Exotic Options / Variance Swaps and Volatility Derivatives

Variance Swaps in the Heston Model — The Effect of Jumps and Dependence on Skew

The model-free variance strike formula prices variance swaps entirely from observable option prices — no model needed. But traders also want to understand why the variance strike has the level it does, and how it would move under a regime change or shock. This requires a model. The Heston stochastic volatility model delivers a closed-form expression for the fair variance strike that is both analytically tractable and financially interpretable — it shows precisely how the variance risk premium depends on the long-run vol, mean-reversion speed, vol-of-vol, and the log-return/vol correlation.
Learning objectives. By the end of this note you should be able to:
  • derive the Heston model's closed-form fair variance strike;
  • interpret the dependence on κ,θ,η,ρ\kappa, \theta, \eta, \rho;
  • describe the jump correction term in the Bates/SVJ extension;
  • compute the variance swap Greeks (sensitivity to Heston parameters).

In this lesson

  1. 3.1 Heston variance dynamics
  2. 3.2 The Heston fair variance strike
  3. 3.3 Jump correction — the Bates model
  4. 3.4 Python: Heston variance swap term structure

Before you start

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